A generalization of commuting probability of finite rings

2018 ◽  
Vol 11 (02) ◽  
pp. 1850023 ◽  
Author(s):  
Parama Dutta ◽  
Rajat Kanti Nath
Keyword(s):  
Commutative Ring ◽  
Finite Rings ◽  
Commuting Probability

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

scholarly journals Some bounds for commuting probability of finite rings

2018 ◽  
Vol 129 (1) ◽  
Author(s):  
Dhiren Kumar Basnet ◽  
Jutirekha Dutta
Keyword(s):  
Finite Rings ◽  
Commuting Probability

Finite Rings Whose Graphs Have Clique Number Less than Five

2021 ◽  
Vol 28 (03) ◽  
pp. 533-540
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Clique Number ◽  
Commutative Rings ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

Residually Finite Rings

1970 ◽  
Vol 22 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Kim Lin Chew ◽  
Sherry Lawn
Keyword(s):  
Commutative Ring ◽  
Homomorphic Image ◽  
Finite Index ◽  
Residue Class ◽  
Finite Rings ◽  
Residually Finite ◽  
Residue Class Ring ◽  
Class Ring ◽  
Equivalent Conditions

Throughout this paper a ring will always be an associative, not necessarily commutative ring with an identity. It is tacitly assumed that the identity of a subring coincides with that of the whole ring. A ring R is said to be residually finite if it satisfies one of the following equivalent conditions:(1) Every non-zero ideal of R is of finite index in R;(2) For each non-zero ideal A of R, the residue class ring R/A is finite;(3) Every proper homomorphic image of R is finite.The class of residually finite rings is large enough to merit our investigation. All finite rings and all simple rings are trivially residually finite. Other residually finite rings are said to be proper.

scholarly journals On commuting probability of finite rings

2017 ◽  
Vol 28 (2) ◽  
pp. 372-382 ◽  
Author(s):  
Jutirekha Dutta ◽  
Dhiren Kumar Basnet ◽  
Rajat Kanti Nath
Keyword(s):  
Finite Rings ◽  
Commuting Probability

scholarly journals On relative commuting probability of finite rings

2019 ◽  
Vol 20 (1) ◽  
pp. 225
Author(s):  
Parama Dutta ◽  
Rajat Kanti Nath
Keyword(s):  
Finite Rings ◽  
Commuting Probability

scholarly journals Digraphs associated with finite rings

2012 ◽  
Vol 92 (106) ◽  
pp. 35-41 ◽  
Author(s):  
Aleksandar Lipkovski
Keyword(s):  
Commutative Ring ◽  
Directed Graph ◽  
Main Idea ◽  
Finite Rings ◽  
Graph Properties ◽  
Finite Commutative Ring

Let A be a finite commutative ring with unity (ring for short). Define a mapping ? : A2 ? A2 by (a, b) 7? (a + b, ab). One can interpret this mapping as a finite directed graph (digraph) G = G(A) with vertices A2 and arrows defined by ?. The main idea is to connect ring properties of A to graph properties of G. Particularly interesting are rings A = Z/nZ. Their graphs should reflect number-theoretic properties of integers. The first few graphs Gn = G(Z/nZ) are drawn and their numerical parameters calculated. From this list, some interesting properties concerning degrees of vertices and presence of loops are noticed and proved.

Rings whose cozero-divisor graph has crosscap number at most two

2017 ◽  
Vol 09 (06) ◽  
pp. 1750074
Author(s):  
A. Mallika ◽  
R. Kala
Keyword(s):  
Commutative Ring ◽  
Planar Graph ◽  
Orientable Surface ◽  
Outer Face ◽  
Finite Rings ◽  
Minimum Integer ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is [Formula: see text], the set of all non-zero and non-unit elements of [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The Crosscap of a graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that the graph can be embedded in the non-orientable surface [Formula: see text]. The planar graph is called [Formula: see text]-outerplanar if removing all the vertices incident on the outer face yields a [Formula: see text]-outerplanar. The Outerplanarity index of a graph [Formula: see text] is the smallest [Formula: see text] such that [Formula: see text] is [Formula: see text]-outerplanar. In this paper, we characterize the class of rings [Formula: see text] (up to isomorphism) for which [Formula: see text]. Further we characterize all finite rings [Formula: see text] (up to isomorphism) for which [Formula: see text] has an outerplanarity index two.

scholarly journals On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

2022 ◽  
Vol 29 (01) ◽  
pp. 167-180
Author(s):  
Mahdi Reza Khorsandi ◽  
Seyed Reza Musawi
Keyword(s):  
Commutative Ring ◽  
Multiplicative Group ◽  
Cayley Graphs ◽  
Artinian Ring ◽  
Finite Rings ◽  
Artinian Rings ◽  
Vertex Set ◽  
Genus Formula ◽  
Multiplicative Subgroup ◽  
Unitary Cayley Graph

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

scholarly journals The Mathematical Safe Problem and Its Solution (Part 2)

2021 ◽  
pp. 16-28
Author(s):  
Sergii Kryvyi ◽  
Hryhorii Hoherchak
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Computer Games ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Finite Rings ◽  
Prime Field ◽  
Linear Diophantine Equations ◽  
Systems Of Linear Equations ◽  
Mathematical Safe

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

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